100 prisoners' names in boxes, against smart warden
$begingroup$
My question, which I do not know the answer to, concerns the classic 100 prisoners and 100 boxes puzzle, summarized below. The solution to this assumes the names are placed in boxes randomly, and gets a ~30% probability of success.
My question is what happens if the warden can choose how to place the names in the boxes, instead of placing them randomly? Now this is a sort of game theory question. The prisoners can no longer use the same strategy, since the warden can guarantee it fails by placing the names in a long cycle. So, my question is,
What is the best strategy for the prisoners, where the goodness of a strategy is measured as its worst case probability of success against all possible warden strategies, including random ones?
As a warm-up which I cannot solve, what is the prisoner's best strategy if the warden randomly places the names in boxes so the resulting permutation is one of the $99!$ possible cycles of length $100$?
Original puzzle:
100 Prisoners and Boxes
A warden explains to 100 prisoners that he will bring them one by one into a room with 100 boxes. Each prisoners' name will be written on a piece of paper, then the slips will be randomly be placed in the boxes, one name per box. When the prisoner is brought into the room, they may open up to 50 boxes. Afterwards, they leave the room without getting to talk to any of the other prisoners, and the warden closes all the opened boxes (and generally resets the room).
The prisoners win only if every prisoner opens the box containing their name. Before the trial begins, the prisoners may confer and decide on a strategy. How can they win with a fairly good probability? You can do much better than the random strategy, with probability $1/2^{100}$.
mathematics strategy probability game-theory
asked 8 mins ago
Mike EarnestMike Earnest
20.9k574213
$endgroup$
add a comment |
$begingroup$
My question, which I do not know the answer to, concerns the classic 100 prisoners and 100 boxes puzzle, summarized below. The solution to this assumes the names are placed in boxes randomly, and gets a ~30% probability of success.
My question is what happens if the warden can choose how to place the names in the boxes, instead of placing them randomly? Now this is a sort of game theory question. The prisoners can no longer use the same strategy, since the warden can guarantee it fails by placing the names in a long cycle. So, my question is,
What is the best strategy for the prisoners, where the goodness of a strategy is measured as its worst case probability of success against all possible warden strategies, including random ones?
As a warm-up which I cannot solve, what is the prisoner's best strategy if the warden randomly places the names in boxes so the resulting permutation is one of the $99!$ possible cycles of length $100$?
Original puzzle:
100 Prisoners and Boxes
A warden explains to 100 prisoners that he will bring them one by one into a room with 100 boxes. Each prisoners' name will be written on a piece of paper, then the slips will be randomly be placed in the boxes, one name per box. When the prisoner is brought into the room, they may open up to 50 boxes. Afterwards, they leave the room without getting to talk to any of the other prisoners, and the warden closes all the opened boxes (and generally resets the room).
The prisoners win only if every prisoner opens the box containing their name. Before the trial begins, the prisoners may confer and decide on a strategy. How can they win with a fairly good probability? You can do much better than the random strategy, with probability $1/2^{100}$.
mathematics strategy probability game-theory
asked 8 mins ago
Mike EarnestMike Earnest
20.9k574213
$endgroup$
add a comment |
$begingroup$
My question, which I do not know the answer to, concerns the classic 100 prisoners and 100 boxes puzzle, summarized below. The solution to this assumes the names are placed in boxes randomly, and gets a ~30% probability of success.
My question is what happens if the warden can choose how to place the names in the boxes, instead of placing them randomly? Now this is a sort of game theory question. The prisoners can no longer use the same strategy, since the warden can guarantee it fails by placing the names in a long cycle. So, my question is,
What is the best strategy for the prisoners, where the goodness of a strategy is measured as its worst case probability of success against all possible warden strategies, including random ones?
As a warm-up which I cannot solve, what is the prisoner's best strategy if the warden randomly places the names in boxes so the resulting permutation is one of the $99!$ possible cycles of length $100$?
Original puzzle:
100 Prisoners and Boxes
A warden explains to 100 prisoners that he will bring them one by one into a room with 100 boxes. Each prisoners' name will be written on a piece of paper, then the slips will be randomly be placed in the boxes, one name per box. When the prisoner is brought into the room, they may open up to 50 boxes. Afterwards, they leave the room without getting to talk to any of the other prisoners, and the warden closes all the opened boxes (and generally resets the room).
The prisoners win only if every prisoner opens the box containing their name. Before the trial begins, the prisoners may confer and decide on a strategy. How can they win with a fairly good probability? You can do much better than the random strategy, with probability $1/2^{100}$.
mathematics strategy probability game-theory
asked 8 mins ago
Mike EarnestMike Earnest
20.9k574213
$endgroup$
My question, which I do not know the answer to, concerns the classic 100 prisoners and 100 boxes puzzle, summarized below. The solution to this assumes the names are placed in boxes randomly, and gets a ~30% probability of success.
My question is what happens if the warden can choose how to place the names in the boxes, instead of placing them randomly? Now this is a sort of game theory question. The prisoners can no longer use the same strategy, since the warden can guarantee it fails by placing the names in a long cycle. So, my question is,
What is the best strategy for the prisoners, where the goodness of a strategy is measured as its worst case probability of success against all possible warden strategies, including random ones?
As a warm-up which I cannot solve, what is the prisoner's best strategy if the warden randomly places the names in boxes so the resulting permutation is one of the $99!$ possible cycles of length $100$?
Original puzzle:
100 Prisoners and Boxes
A warden explains to 100 prisoners that he will bring them one by one into a room with 100 boxes. Each prisoners' name will be written on a piece of paper, then the slips will be randomly be placed in the boxes, one name per box. When the prisoner is brought into the room, they may open up to 50 boxes. Afterwards, they leave the room without getting to talk to any of the other prisoners, and the warden closes all the opened boxes (and generally resets the room).
The prisoners win only if every prisoner opens the box containing their name. Before the trial begins, the prisoners may confer and decide on a strategy. How can they win with a fairly good probability? You can do much better than the random strategy, with probability $1/2^{100}$.
mathematics strategy probability game-theory
mathematics strategy probability game-theory
asked 8 mins ago
Mike EarnestMike Earnest
20.9k574213
asked 8 mins ago
Mike EarnestMike Earnest
20.9k574213
asked 8 mins ago
Mike EarnestMike Earnest
20.9k574213
asked 8 mins ago
Mike EarnestMike Earnest
20.9k574213
asked 8 mins ago
Mike EarnestMike Earnest
20.9k574213
20.9k574213
add a comment |
add a comment |
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